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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Approximating discrete valuation rings by regular local rings

Author(s): William Heinzer; Christel Rotthaus; Sylvia Wiegand
Journal: Proc. Amer. Math. Soc. 129 (2001), 37-43.
MSC (1991): Primary 13F30, 13H05; Secondary 13E05, 13G05, 13J05, 13J15
Posted: July 27, 2000
MathSciNet review: 1694346
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Abstract | References | Similar articles | Additional information

Abstract: Let $k$ be a field of characteristic zero and let $(V,\mathbf{n})$ be a discrete rank-one valuation domain containing $k$ with $V/\mathbf{n}= k$. Assume that the fraction field $L$ of $V$has finite transcendence degree $s$ over $k$. For every positive integer $d \le s$, we prove that $V$ can be realized as a directed union of regular local $k$-subalgebras of $V$ of dimension $d$.

References:

[A]
S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321-348. MR 18:556b

[AH]
S. Abhyankar and W. Heinzer, Ramification in infinite integral extensions, J. Algebra 170 (1994), 861-879. MR 95k:13006

[C1]
S. Cutkosky, Local factorization of birational maps, Adv. Math. 132 (1997), 167-315. MR 99c:14018

[C2]
-, Local factorization and monomialization of morphisms, to appear.

[HRS]
W. Heinzer, C. Rotthaus and J. Sally, Formal fibers and birational extensions, Nagoya Math J. 131 (1993), 1-38. MR 95a:13008

[HRW1]
W. Heinzer, C. Rotthaus and S. Wiegand, Noetherian rings between a semilocal domain and its completion, J. Algebra 198 (1997), 627-655. MR 99a:13008

[HRW2]
-, Building Noetherian domains inside an ideal-adic completion, Abelian Groups, Module Theory and Topology Proceedings in Honor of Adalberto Orsatti's 60th Birthday (D. Dikranjan and L. Salce, ed.), Dekker Inc, 1998, pp. 279-287. MR 99m:13035

[HRW3]
-, Noetherian domains inside a homomorphic image of a completion, J. Algebra 215 (1999), 666-681. CMP 99:12

[M1]
H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986. MR 88h:13001

[M2]
-, Commutative Algebra, second edition, Benjamin/Cummings, Reading, MA, 1980. MR 82i:13003

[N1]
M. Nagata, An example of a normal local ring which is analytically reducible, Mem. Coll. Sci., Univ. Kyoto 31 (1958), 83-85. MR 20:3864

[N2]
-, Local Rings, John Wiley, 1962. MR 27:5790

[R]
C. Rotthaus, Homomorphic images of regular local rings, Comm. in Alg. 24 (1996), 445-476. MR 97b:13029

[S]
D. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 284-320. MR 48:8492

[Z1]
O. Zariski, Local uniformization on algebraic varieties, Ann. Math. 41 (1940), 852-896. MR 2:124a

[Z2]
-, Applicazioni geometriche della teoria delle valutazioni, Rend. Circolo Mat. Palermo 13(5) (1954), 1-38. MR 16:165f

[ZS]
O. Zariski and P. Samuel, Commutative Algebra vol II, Van Nostrand, Princeton, NJ, 1960. MR 22:11006

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Additional Information:

William Heinzer
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395
Email: heinzer@math.purdue.edu

Christel Rotthaus
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: rotthaus@math.msu.edu

Sylvia Wiegand
Affiliation: Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323
Email: swiegand@math.unl.edu

DOI: 10.1090/S0002-9939-00-05492-7
PII: S 0002-9939(00)05492-7
Keywords: Discrete rank-one valuation domain, \'{e}tale extension, excellent ring, Henselization, local uniformization, regular local domain
Received by editor(s): July 23, 1998
Received by editor(s) in revised form: March 22, 1999
Posted: July 27, 2000
Additional Notes: The authors thank the National Science Foundation and the National Security Agency for support for this research. In addition they are grateful for the hospitality and cooperation of Michigan State University, the University of Nebraska and Purdue University, where several work sessions on this research were conducted.
Communicated by: Wolmer V. Vasconcelos
Copyright of article: Copyright 2000, American Mathematical Society




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